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## G = SD32⋊3D5order 320 = 26·5

### The semidirect product of SD32 and D5 acting through Inn(SD32)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — SD32⋊3D5
 Chief series C1 — C5 — C10 — C20 — C40 — C8×D5 — D8⋊3D5 — SD32⋊3D5
 Lower central C5 — C10 — C20 — C40 — SD32⋊3D5
 Upper central C1 — C2 — C4 — C8 — SD32

Generators and relations for SD323D5
G = < a,b,c,d | a16=b2=c5=d2=1, bab=a7, ac=ca, ad=da, bc=cb, dbd=a8b, dcd=c-1 >

Subgroups: 422 in 84 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2 [×3], C4, C4 [×3], C22 [×3], C5, C8, C8, C2×C4 [×3], D4 [×4], Q8 [×2], D5 [×2], C10, C10, C16, C16, C2×C8, D8, D8, SD16 [×2], Q16, Q16, C4○D4 [×2], Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C16, D16, SD32, SD32, Q32, C4○D8 [×2], C52C8, C40, Dic10, C4×D5, C4×D5, D20 [×2], C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C4○D16, C52C16, C80, C8×D5, D40, Dic20, D4.D5, Q8⋊D5, C5×D8, C5×Q16, D42D5, Q82D5, D5×C16, C16⋊D5, C5⋊D16, C5⋊Q32, C5×SD32, D83D5, Q8.D10, SD323D5
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, D10 [×3], C2×D8, C22×D5, C4○D16, D4×D5, D5×D8, SD323D5

Smallest permutation representation of SD323D5
On 160 points
Generators in S160
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)
(1 67)(2 74)(3 65)(4 72)(5 79)(6 70)(7 77)(8 68)(9 75)(10 66)(11 73)(12 80)(13 71)(14 78)(15 69)(16 76)(17 55)(18 62)(19 53)(20 60)(21 51)(22 58)(23 49)(24 56)(25 63)(26 54)(27 61)(28 52)(29 59)(30 50)(31 57)(32 64)(33 115)(34 122)(35 113)(36 120)(37 127)(38 118)(39 125)(40 116)(41 123)(42 114)(43 121)(44 128)(45 119)(46 126)(47 117)(48 124)(81 109)(82 100)(83 107)(84 98)(85 105)(86 112)(87 103)(88 110)(89 101)(90 108)(91 99)(92 106)(93 97)(94 104)(95 111)(96 102)(129 145)(130 152)(131 159)(132 150)(133 157)(134 148)(135 155)(136 146)(137 153)(138 160)(139 151)(140 158)(141 149)(142 156)(143 147)(144 154)
(1 37 17 151 83)(2 38 18 152 84)(3 39 19 153 85)(4 40 20 154 86)(5 41 21 155 87)(6 42 22 156 88)(7 43 23 157 89)(8 44 24 158 90)(9 45 25 159 91)(10 46 26 160 92)(11 47 27 145 93)(12 48 28 146 94)(13 33 29 147 95)(14 34 30 148 96)(15 35 31 149 81)(16 36 32 150 82)(49 133 101 77 121)(50 134 102 78 122)(51 135 103 79 123)(52 136 104 80 124)(53 137 105 65 125)(54 138 106 66 126)(55 139 107 67 127)(56 140 108 68 128)(57 141 109 69 113)(58 142 110 70 114)(59 143 111 71 115)(60 144 112 72 116)(61 129 97 73 117)(62 130 98 74 118)(63 131 99 75 119)(64 132 100 76 120)
(1 106)(2 107)(3 108)(4 109)(5 110)(6 111)(7 112)(8 97)(9 98)(10 99)(11 100)(12 101)(13 102)(14 103)(15 104)(16 105)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 63)(27 64)(28 49)(29 50)(30 51)(31 52)(32 53)(33 134)(34 135)(35 136)(36 137)(37 138)(38 139)(39 140)(40 141)(41 142)(42 143)(43 144)(44 129)(45 130)(46 131)(47 132)(48 133)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)(71 88)(72 89)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 81)(113 154)(114 155)(115 156)(116 157)(117 158)(118 159)(119 160)(120 145)(121 146)(122 147)(123 148)(124 149)(125 150)(126 151)(127 152)(128 153)

G:=sub<Sym(160)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,67)(2,74)(3,65)(4,72)(5,79)(6,70)(7,77)(8,68)(9,75)(10,66)(11,73)(12,80)(13,71)(14,78)(15,69)(16,76)(17,55)(18,62)(19,53)(20,60)(21,51)(22,58)(23,49)(24,56)(25,63)(26,54)(27,61)(28,52)(29,59)(30,50)(31,57)(32,64)(33,115)(34,122)(35,113)(36,120)(37,127)(38,118)(39,125)(40,116)(41,123)(42,114)(43,121)(44,128)(45,119)(46,126)(47,117)(48,124)(81,109)(82,100)(83,107)(84,98)(85,105)(86,112)(87,103)(88,110)(89,101)(90,108)(91,99)(92,106)(93,97)(94,104)(95,111)(96,102)(129,145)(130,152)(131,159)(132,150)(133,157)(134,148)(135,155)(136,146)(137,153)(138,160)(139,151)(140,158)(141,149)(142,156)(143,147)(144,154), (1,37,17,151,83)(2,38,18,152,84)(3,39,19,153,85)(4,40,20,154,86)(5,41,21,155,87)(6,42,22,156,88)(7,43,23,157,89)(8,44,24,158,90)(9,45,25,159,91)(10,46,26,160,92)(11,47,27,145,93)(12,48,28,146,94)(13,33,29,147,95)(14,34,30,148,96)(15,35,31,149,81)(16,36,32,150,82)(49,133,101,77,121)(50,134,102,78,122)(51,135,103,79,123)(52,136,104,80,124)(53,137,105,65,125)(54,138,106,66,126)(55,139,107,67,127)(56,140,108,68,128)(57,141,109,69,113)(58,142,110,70,114)(59,143,111,71,115)(60,144,112,72,116)(61,129,97,73,117)(62,130,98,74,118)(63,131,99,75,119)(64,132,100,76,120), (1,106)(2,107)(3,108)(4,109)(5,110)(6,111)(7,112)(8,97)(9,98)(10,99)(11,100)(12,101)(13,102)(14,103)(15,104)(16,105)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,49)(29,50)(30,51)(31,52)(32,53)(33,134)(34,135)(35,136)(36,137)(37,138)(38,139)(39,140)(40,141)(41,142)(42,143)(43,144)(44,129)(45,130)(46,131)(47,132)(48,133)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(71,88)(72,89)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,81)(113,154)(114,155)(115,156)(116,157)(117,158)(118,159)(119,160)(120,145)(121,146)(122,147)(123,148)(124,149)(125,150)(126,151)(127,152)(128,153)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,67)(2,74)(3,65)(4,72)(5,79)(6,70)(7,77)(8,68)(9,75)(10,66)(11,73)(12,80)(13,71)(14,78)(15,69)(16,76)(17,55)(18,62)(19,53)(20,60)(21,51)(22,58)(23,49)(24,56)(25,63)(26,54)(27,61)(28,52)(29,59)(30,50)(31,57)(32,64)(33,115)(34,122)(35,113)(36,120)(37,127)(38,118)(39,125)(40,116)(41,123)(42,114)(43,121)(44,128)(45,119)(46,126)(47,117)(48,124)(81,109)(82,100)(83,107)(84,98)(85,105)(86,112)(87,103)(88,110)(89,101)(90,108)(91,99)(92,106)(93,97)(94,104)(95,111)(96,102)(129,145)(130,152)(131,159)(132,150)(133,157)(134,148)(135,155)(136,146)(137,153)(138,160)(139,151)(140,158)(141,149)(142,156)(143,147)(144,154), (1,37,17,151,83)(2,38,18,152,84)(3,39,19,153,85)(4,40,20,154,86)(5,41,21,155,87)(6,42,22,156,88)(7,43,23,157,89)(8,44,24,158,90)(9,45,25,159,91)(10,46,26,160,92)(11,47,27,145,93)(12,48,28,146,94)(13,33,29,147,95)(14,34,30,148,96)(15,35,31,149,81)(16,36,32,150,82)(49,133,101,77,121)(50,134,102,78,122)(51,135,103,79,123)(52,136,104,80,124)(53,137,105,65,125)(54,138,106,66,126)(55,139,107,67,127)(56,140,108,68,128)(57,141,109,69,113)(58,142,110,70,114)(59,143,111,71,115)(60,144,112,72,116)(61,129,97,73,117)(62,130,98,74,118)(63,131,99,75,119)(64,132,100,76,120), (1,106)(2,107)(3,108)(4,109)(5,110)(6,111)(7,112)(8,97)(9,98)(10,99)(11,100)(12,101)(13,102)(14,103)(15,104)(16,105)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,49)(29,50)(30,51)(31,52)(32,53)(33,134)(34,135)(35,136)(36,137)(37,138)(38,139)(39,140)(40,141)(41,142)(42,143)(43,144)(44,129)(45,130)(46,131)(47,132)(48,133)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(71,88)(72,89)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,81)(113,154)(114,155)(115,156)(116,157)(117,158)(118,159)(119,160)(120,145)(121,146)(122,147)(123,148)(124,149)(125,150)(126,151)(127,152)(128,153) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)], [(1,67),(2,74),(3,65),(4,72),(5,79),(6,70),(7,77),(8,68),(9,75),(10,66),(11,73),(12,80),(13,71),(14,78),(15,69),(16,76),(17,55),(18,62),(19,53),(20,60),(21,51),(22,58),(23,49),(24,56),(25,63),(26,54),(27,61),(28,52),(29,59),(30,50),(31,57),(32,64),(33,115),(34,122),(35,113),(36,120),(37,127),(38,118),(39,125),(40,116),(41,123),(42,114),(43,121),(44,128),(45,119),(46,126),(47,117),(48,124),(81,109),(82,100),(83,107),(84,98),(85,105),(86,112),(87,103),(88,110),(89,101),(90,108),(91,99),(92,106),(93,97),(94,104),(95,111),(96,102),(129,145),(130,152),(131,159),(132,150),(133,157),(134,148),(135,155),(136,146),(137,153),(138,160),(139,151),(140,158),(141,149),(142,156),(143,147),(144,154)], [(1,37,17,151,83),(2,38,18,152,84),(3,39,19,153,85),(4,40,20,154,86),(5,41,21,155,87),(6,42,22,156,88),(7,43,23,157,89),(8,44,24,158,90),(9,45,25,159,91),(10,46,26,160,92),(11,47,27,145,93),(12,48,28,146,94),(13,33,29,147,95),(14,34,30,148,96),(15,35,31,149,81),(16,36,32,150,82),(49,133,101,77,121),(50,134,102,78,122),(51,135,103,79,123),(52,136,104,80,124),(53,137,105,65,125),(54,138,106,66,126),(55,139,107,67,127),(56,140,108,68,128),(57,141,109,69,113),(58,142,110,70,114),(59,143,111,71,115),(60,144,112,72,116),(61,129,97,73,117),(62,130,98,74,118),(63,131,99,75,119),(64,132,100,76,120)], [(1,106),(2,107),(3,108),(4,109),(5,110),(6,111),(7,112),(8,97),(9,98),(10,99),(11,100),(12,101),(13,102),(14,103),(15,104),(16,105),(17,54),(18,55),(19,56),(20,57),(21,58),(22,59),(23,60),(24,61),(25,62),(26,63),(27,64),(28,49),(29,50),(30,51),(31,52),(32,53),(33,134),(34,135),(35,136),(36,137),(37,138),(38,139),(39,140),(40,141),(41,142),(42,143),(43,144),(44,129),(45,130),(46,131),(47,132),(48,133),(65,82),(66,83),(67,84),(68,85),(69,86),(70,87),(71,88),(72,89),(73,90),(74,91),(75,92),(76,93),(77,94),(78,95),(79,96),(80,81),(113,154),(114,155),(115,156),(116,157),(117,158),(118,159),(119,160),(120,145),(121,146),(122,147),(123,148),(124,149),(125,150),(126,151),(127,152),(128,153)])

44 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 8A 8B 8C 8D 10A 10B 10C 10D 16A 16B 16C 16D 16E 16F 16G 16H 20A 20B 20C 20D 40A 40B 40C 40D 80A ··· 80H order 1 2 2 2 2 4 4 4 4 4 5 5 8 8 8 8 10 10 10 10 16 16 16 16 16 16 16 16 20 20 20 20 40 40 40 40 80 ··· 80 size 1 1 8 10 40 2 5 5 8 40 2 2 2 2 10 10 2 2 16 16 2 2 2 2 10 10 10 10 4 4 16 16 4 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 D8 D8 D10 D10 D10 C4○D16 D4×D5 D5×D8 SD32⋊3D5 kernel SD32⋊3D5 D5×C16 C16⋊D5 C5⋊D16 C5⋊Q32 C5×SD32 D8⋊3D5 Q8.D10 C5⋊2C8 C4×D5 SD32 Dic5 D10 C16 D8 Q16 C5 C4 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 8 2 4 8

Matrix representation of SD323D5 in GL4(𝔽241) generated by

 138 41 0 0 200 138 0 0 0 0 1 0 0 0 0 1
,
 214 156 0 0 156 27 0 0 0 0 240 0 0 0 0 240
,
 1 0 0 0 0 1 0 0 0 0 51 1 0 0 240 0
,
 0 177 0 0 64 0 0 0 0 0 240 190 0 0 0 1
G:=sub<GL(4,GF(241))| [138,200,0,0,41,138,0,0,0,0,1,0,0,0,0,1],[214,156,0,0,156,27,0,0,0,0,240,0,0,0,0,240],[1,0,0,0,0,1,0,0,0,0,51,240,0,0,1,0],[0,64,0,0,177,0,0,0,0,0,240,0,0,0,190,1] >;

SD323D5 in GAP, Magma, Sage, TeX

{\rm SD}_{32}\rtimes_3D_5
% in TeX

G:=Group("SD32:3D5");
// GroupNames label

G:=SmallGroup(320,543);
// by ID

G=gap.SmallGroup(320,543);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,758,135,184,346,185,192,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^16=b^2=c^5=d^2=1,b*a*b=a^7,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^8*b,d*c*d=c^-1>;
// generators/relations

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